## Abstract

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the B_{k}-VPG graphs, k ≥ 0. In chip manufacturing, circuit layout is modeled as paths (wires) on a grid, where it is natural to constrain the number of bends per wire for reasons of feasibility and to reduce the cost of the chip. If the number k of bends is not restricted, then the VPG graphs are equiv-alent to the well-known class of string graphs, namely, the intersection graphs of arbitrary curves in the plane. In the case of B_{0}-VPG graphs, we observe that horizontal and vertical segments have strong Helly number 2, and thus the clique problem has polynomial-time complexity, given the path representation. The recognition and coloring problems for B_{0}-VPG graphs, however, are NP-complete. We give a 2-approximation algorithm for coloring B_{0}-VPG graphs. Furthermore, we prove that triangle-free B_{0}-VPG graphs are 4-colorable, and this is best possible. We present a hierarchy of VPG graphs relating them to other known families of graphs. The grid intersection graphs are shown to be equivalent to the bi-partite B_{0}-VPG graphs and the circle graphs are strictly contained in B_{1}-VPG. We prove the strict containment of B_{0}-VPG into B_{1}-VPG, and we conjecture that, in general, this strict containment continues for all values of k. We present a graph which is not in B_{1}-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B_{3}-VPG graphs, although it is not known if this is best possible.

Original language | English |
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Pages (from-to) | 129-150 |

Number of pages | 22 |

Journal | Journal of Graph Algorithms and Applications |

Volume | 16 |

Issue number | 2 |

DOIs | |

State | Published - 2012 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science
- Computer Science Applications
- Geometry and Topology
- Computational Theory and Mathematics